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Theorem sstpr 3519
Description: The subsets of a triple. (Contributed by Jim Kingdon, 11-Aug-2018.)
Assertion
Ref Expression
sstpr ((((A = ∅ A = {B}) (A = {𝐶} A = {B, 𝐶})) ((A = {𝐷} A = {B, 𝐷}) (A = {𝐶, 𝐷} A = {B, 𝐶, 𝐷}))) → A ⊆ {B, 𝐶, 𝐷})

Proof of Theorem sstpr
StepHypRef Expression
1 ssprr 3518 . . 3 (((A = ∅ A = {B}) (A = {𝐶} A = {B, 𝐶})) → A ⊆ {B, 𝐶})
2 prsstp12 3508 . . 3 {B, 𝐶} ⊆ {B, 𝐶, 𝐷}
31, 2syl6ss 2951 . 2 (((A = ∅ A = {B}) (A = {𝐶} A = {B, 𝐶})) → A ⊆ {B, 𝐶, 𝐷})
4 snsstp3 3507 . . . . 5 {𝐷} ⊆ {B, 𝐶, 𝐷}
5 sseq1 2960 . . . . 5 (A = {𝐷} → (A ⊆ {B, 𝐶, 𝐷} ↔ {𝐷} ⊆ {B, 𝐶, 𝐷}))
64, 5mpbiri 157 . . . 4 (A = {𝐷} → A ⊆ {B, 𝐶, 𝐷})
7 prsstp13 3509 . . . . 5 {B, 𝐷} ⊆ {B, 𝐶, 𝐷}
8 sseq1 2960 . . . . 5 (A = {B, 𝐷} → (A ⊆ {B, 𝐶, 𝐷} ↔ {B, 𝐷} ⊆ {B, 𝐶, 𝐷}))
97, 8mpbiri 157 . . . 4 (A = {B, 𝐷} → A ⊆ {B, 𝐶, 𝐷})
106, 9jaoi 635 . . 3 ((A = {𝐷} A = {B, 𝐷}) → A ⊆ {B, 𝐶, 𝐷})
11 prsstp23 3510 . . . . 5 {𝐶, 𝐷} ⊆ {B, 𝐶, 𝐷}
12 sseq1 2960 . . . . 5 (A = {𝐶, 𝐷} → (A ⊆ {B, 𝐶, 𝐷} ↔ {𝐶, 𝐷} ⊆ {B, 𝐶, 𝐷}))
1311, 12mpbiri 157 . . . 4 (A = {𝐶, 𝐷} → A ⊆ {B, 𝐶, 𝐷})
14 eqimss 2991 . . . 4 (A = {B, 𝐶, 𝐷} → A ⊆ {B, 𝐶, 𝐷})
1513, 14jaoi 635 . . 3 ((A = {𝐶, 𝐷} A = {B, 𝐶, 𝐷}) → A ⊆ {B, 𝐶, 𝐷})
1610, 15jaoi 635 . 2 (((A = {𝐷} A = {B, 𝐷}) (A = {𝐶, 𝐷} A = {B, 𝐶, 𝐷})) → A ⊆ {B, 𝐶, 𝐷})
173, 16jaoi 635 1 ((((A = ∅ A = {B}) (A = {𝐶} A = {B, 𝐶})) ((A = {𝐷} A = {B, 𝐷}) (A = {𝐶, 𝐷} A = {B, 𝐶, 𝐷}))) → A ⊆ {B, 𝐶, 𝐷})
Colors of variables: wff set class
Syntax hints:  wi 4   wo 628   = wceq 1242  wss 2911  c0 3218  {csn 3367  {cpr 3368  {ctp 3369
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3or 885  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-sn 3373  df-pr 3374  df-tp 3375
This theorem is referenced by:  pwtpss  3568
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